Now this could look deceptively simple if we didn't use operators for energy and momentum. We could simply divide by the wave function Ψ. But like most things, it's never simple. The energy operator acts on the wave function, as does the momentum operator. So we need to find the wave function in order to make any sense of this equation.

In Quantum Mechanics, everything is probabilistic (e.g., the probability of finding a particle is the square of the amplitude of the wave function). So we often want to know the expected value of position, momentum, or anything else, and there is quite a nice method of doing this.

This notation is very widely used. It is extremely useful in finite dimensional problems, represented by matrices and state vectors. In 2D, qubits are used (most used in Quantum Information Theory), where:

Commutators are very important in Quantum Mechanics. As well as being how Heisenberg discovered the Uncertainty Principle, they are often used in particle physics. It is known that you cannot know the value of two physical values at the same time if they do not commute.